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Theorem iscmgmALT 44527
 Description: The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b 𝐵 = (Base‘𝑀)
ismgmALT.o = (+g𝑀)
Assertion
Ref Expression
iscmgmALT (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))

Proof of Theorem iscmgmALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6646 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6646 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 5044 . . 3 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ (+g𝑀) comLaw (Base‘𝑀)))
4 ismgmALT.o . . . 4 = (+g𝑀)
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5breq12i 5040 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6bitr4di 292 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-cmgm2 44523 . 2 CMgmALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3631 1 (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5031  ‘cfv 6325  Basecbs 16478  +gcplusg 16560   comLaw ccomlaw 44488  MgmALTcmgm2 44518  CMgmALTccmgm2 44519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-cmgm2 44523 This theorem is referenced by: (None)
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